Meander (mathematics)

In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges.

Formal definition of meander

Given a fixed oriented line L in the Euclidean plane R², a meander of order n is a non-self-intersecting closed curve in R² which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

The number of distinct meanders of order n is the meandric number M_n. The first fifteen meandric numbers are given below (for more terms, see OEIS A005315).

M₁ = 1

M₂ = 2

M₃ = 8

M₄ = 42

M₅ = 262

M₆ = 1828

M₇ = 13820

M₈ = 110954

M₉ = 933458

M₁₀ = 8152860

M₁₁ = 73424650

M₁₂ = 678390116

M₁₃ = 6405031050

M₁₄ = 61606881612

M₁₅ = 602188541928

Examples of meanders

The meander of order 1 intersects the line twice:

The meanders of order 2 intersect the line four times:

File:Meander M21 jaredwf.png File:Meander M22 jaredwf.png

Formal definition of open meander

Given a fixed oriented line L in the Euclidean plane R², an open meander of order n is a non-self-intersecting oriented curve in R² which transversally intersects the line at n points for some positive integer n. Two open meanders are said to be equivalent if they are homeomorphic in the plane.

The number of distinct open meanders of order n is the open meandric number m_n. The first fifteen open meandric numbers are given below (for more terms, see OEIS A005316).

m₁ = 1

m₂ = 1

m₃ = 2

m₄ = 3

m₅ = 8

m₆ = 14

m₇ = 42

m₈ = 81

m₉ = 262

m₁₀ = 538

m₁₁ = 1828

m₁₂ = 3926

m₁₃ = 13820

m₁₄ = 30694

m₁₅ = 110954

Examples of open meanders

The open meander of order 1 intersects the line once:

The open meander of order 2 intersects the line twice:

Formal definition of semi-meander

Given a fixed oriented ray R in the Euclidean plane R², a semi-meander of order n is a non-self-intersecting closed curve in R² which transversally intersects the ray at n points for some positive integer n. Two semi-meanders are said to be equivalent if they are homeomorphic in the plane.

The number of distinct semi-meanders of order n is the semi-meandric number M_n (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below (for more terms, see OEIS A000682).

M₁ = 1

M₂ = 1

M₃ = 2

M₄ = 4

M₅ = 10

M₆ = 24

M₇ = 66

M₈ = 174

M₉ = 504

M₁₀ = 1406

M₁₁ = 4210

M₁₂ = 12198

M₁₃ = 37378

M₁₄ = 111278

M₁₅ = 346846

Examples of semi-meanders

The semi-meander of order 1 intersects the ray once:

File:Semi-meander M1 jaredwf.png

The semi-meander of order 2 intersects the ray twice:

Properties of meanderic numbers

There is a one-to-one correspondence between meandric and open meandric numbers:

M_n = m_2n-1

Each meandric number can be bounded by semi-meandric numbers:

M_n ≤ M_n ≤ M_2n

For n > 1, meanderic numbers are even:

M_n ≡ 0 (mod 2)